Its hard for me to have a favorite faction, but I'll share some thoughts:

Carthage has a excellent variety of cavalry. They have every sort of cavalry except horse archers

As you said, the Steppe factions are an obvious choice, but, since most of their units are armed with bows, it can be tricky to get their horsemen to charge with melee weapons instead of bows.

I haven't played enough of the Successors to have an opinion

Hayasdan has very powerful family members and elite cavalry, but their horse units are very expensive to maintain. With my empire, I have a band of "enforcers" made up mostly of Scythian riders and kinsman heavy cavalry. Good for trampling rebel stacks and the more pathetic stacks to AI send to besiege my cities. Right now, my biggest stack is fighting hordes of Saba lead Nubian light spearmen.

Saba has access to Arab Light Cavalry and some decent medium cavalry. The former is suprisingly effective. I've been using lots of them to disrupt Hellenic lines by running right behind their phalanxes and pelting their backs. The fights can be very hard, but very satisfying. It is fun to see them swarm over elite Ptolemaic units The Arab Light Cavalry have AP lances that they can crouch, so if properly used, a charge from all sides can break almost anything. However, Saba has foot bound family members.

Gallic Light Cavalry is an excellent jack-of-all-trades cavalry.